| L(s) = 1 | − 0.601·2-s − 1.63·4-s − 5-s − 2.59·7-s + 2.18·8-s + 0.601·10-s − 5.12·11-s + 13-s + 1.56·14-s + 1.96·16-s − 5.36·17-s − 4.53·19-s + 1.63·20-s + 3.08·22-s + 0.799·23-s + 25-s − 0.601·26-s + 4.25·28-s − 3.33·29-s − 6.98·31-s − 5.55·32-s + 3.23·34-s + 2.59·35-s − 10.6·37-s + 2.72·38-s − 2.18·40-s + 4.53·41-s + ⋯ |
| L(s) = 1 | − 0.425·2-s − 0.819·4-s − 0.447·5-s − 0.982·7-s + 0.773·8-s + 0.190·10-s − 1.54·11-s + 0.277·13-s + 0.417·14-s + 0.490·16-s − 1.30·17-s − 1.04·19-s + 0.366·20-s + 0.656·22-s + 0.166·23-s + 0.200·25-s − 0.117·26-s + 0.804·28-s − 0.618·29-s − 1.25·31-s − 0.982·32-s + 0.553·34-s + 0.439·35-s − 1.74·37-s + 0.442·38-s − 0.346·40-s + 0.708·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03968298351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03968298351\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 0.601T + 2T^{2} \) |
| 7 | \( 1 + 2.59T + 7T^{2} \) |
| 11 | \( 1 + 5.12T + 11T^{2} \) |
| 17 | \( 1 + 5.36T + 17T^{2} \) |
| 19 | \( 1 + 4.53T + 19T^{2} \) |
| 23 | \( 1 - 0.799T + 23T^{2} \) |
| 29 | \( 1 + 3.33T + 29T^{2} \) |
| 31 | \( 1 + 6.98T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 4.53T + 41T^{2} \) |
| 43 | \( 1 - 6.65T + 43T^{2} \) |
| 47 | \( 1 + 5.35T + 47T^{2} \) |
| 53 | \( 1 + 9.68T + 53T^{2} \) |
| 59 | \( 1 + 0.818T + 59T^{2} \) |
| 61 | \( 1 + 4.24T + 61T^{2} \) |
| 67 | \( 1 + 9.64T + 67T^{2} \) |
| 71 | \( 1 + 1.13T + 71T^{2} \) |
| 73 | \( 1 - 1.97T + 73T^{2} \) |
| 79 | \( 1 - 7.27T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 - 4.74T + 89T^{2} \) |
| 97 | \( 1 + 6.96T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.268353082562202336795250081273, −7.60656693163728046068710797326, −6.92072606393549041456888609525, −6.04326996771274999162026365846, −5.22525958723803182008300363936, −4.49106820980911876556588985932, −3.74443196364539134297505736106, −2.90119251076059206740012852570, −1.82418905747290722540726431528, −0.10982168765184806457121626049,
0.10982168765184806457121626049, 1.82418905747290722540726431528, 2.90119251076059206740012852570, 3.74443196364539134297505736106, 4.49106820980911876556588985932, 5.22525958723803182008300363936, 6.04326996771274999162026365846, 6.92072606393549041456888609525, 7.60656693163728046068710797326, 8.268353082562202336795250081273
